The Discrete Approach and Boundary Behavior

The path integral or the Wiener measure interpretation of stochastic differential equations is useful for both the conceptual understanding of stochastic differential equations and for deriving differential equations that govern the evolution of the pdfs of their solutions. A simple illustration of the computational usefulness of the Wiener measure is the easy derivation of the explicit expression (2.25) for the pdf of the MBM. Unfortunately, no explicit expressions exist in general for the pdf of the solution to (5.1). The second best to such an explicit expression is a (deterministic) differential equation for the pdf, whose solution can be studied both analytically and numerically directly from the differential equation. A case in point is the diffusion equation and the initial condition (2.26) that the pdf of the MBM satisfies.